Actually, I think you bring up some really good points. If you believe the reals are countable, then your task is to come up with one, just one, scheme for enu

Actually, I think you bring up some really good points. If you believe the reals are countable, then your task is to come up with one, just one, scheme for enumerating them (i.e. putting in 1-to-1 correspondence with the naturals). If, on the other hand, you believe the reals are uncountable, then your task is decidedly harder: You need to come up with some scheme that, given any proposed enumeration, you can produce a number definitively not on that list. Cantor's argument does the latter. Given any scheme---say a computer program---that claims to enumerate the reals, Cantor's argument shows us how to build a corresponding computer program to generate the digits of a number that are never produced by your original program. I.e. if you let the original program list out N reals, Cantor's program can give you an N-digit number that it has definitely not listed. The original program certainly might list it later, but by that point it's listed N+M digits and Cantor's program can give an N+M-digit number that's missing. The point being that Cantor's program always works, no matter how big N gets. On the other hand, as another poster has pointed out, the Cantor scheme fails to work for an actually countable set, like the even numbers or primes. Cantor's argument will still work in a sense. It will give you something, but the thing it gives you will always violate some property that the original set of things must satisfy, like having only finite digits or whatnot.